Physical Foundations of Cosmology

348 Inflation II: origin of the primordial inhomogeneities

no preferred position in space? Quantum mechanical unitary evolution does not de-
stroy translational invariance and hence the answer to this question must lie in the
transition from quantum fluctuations to classical inhomogeneities. Decoherence is
a necessary condition for the emergence of classical inhomogeneities and can easily
be justified for amplified cosmological perturbations. However, decoherence is not
sufficient to explain the breaking of translational invariance. It can be shown that
as a result of unitary evolution we obtain a state which is a superposition of many
macroscopically different states, each corresponding to a particular realization of
galaxy distribution. Most of these realizations have the same statistical properties.
Such a state is a close cosmic analog of the “Schr ̈odinger cat.” Therefore, to pick
an observed macroscopic state from the superposition we have to appeal either to
Bohr’s reduction postulate or to Everett’s many-worlds interpretation of quantum
mechanics. The first possibility does not look convincing in the cosmological con-
text. The reader who would like to pursue this issue can consult the corresponding
references in “Bibliography” (Everett, 1957; De Witt and Graham, 1973).

8.4 Gravitational waves from inflation

Quantizing gravitational wavesIn a similar manner to scalar perturbations, long-
wavelength gravitational waves are also generated in inflation. The calculations are
not very different from those presented in the previous section. First of all we need
the action for the gravitational waves. This action can be derived by expanding the
Einstein action up to the second order in transverse, traceless metric perturbations
hik. The result is

S=

1

64 π

∫

a^2

(

hij′hij′−hij,lhij,l

)

dηd^3 x, (8.111)

where the spatial indices are raised and lowered with the help of the unit tensorδik.

Problem 8.15Derive (8.111). (HintCalculate the curvature tensorRconsidering
small perturbations around Minkowski space and then use (5.111) to make the
appropriate conformal transformation to an expanding universe.)

Substituting the expansion

hij(x,η)=

∫

hk(η)eij(k)eikx

d^3 k

( 2 π)^3 /^2

, (8.112)

whereeij(k)is the polarization tensor, into (8.111), we obtain

S=

1

64 π

∫

a^2 eijeij

(

h′kh′−k−k^2 hkh−k

)

dηd^3 k. (8.113)